Urcohomologies and cohomologies of $N$-complexes

Abstract

For a general sequence of objects and morphisms, we construct two $N$-complexes. Then we can define cohomologies $(i,k)$-type of the $N$-complexes not only on a diagonal region but also in the triangular region. We obtain an invariant defined on a general sequence of objects and morphisms. For a short exact sequence of $N$-complexes, we get the associated long exact sequence generalizing the classical long exact sequence. Lastly, several properties of the vanishing cohomologies of $N$-complexes are given.